let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 holds
( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 holds
( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 implies ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) )
assume that
A1: Z c= dom ((1 / (4 * a)) (#) (sin * f)) and
A2: ( ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 ) ; :: thesis: ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) )
A3: Z c= dom (sin * f) by A1, VALUED_1:def 5;
A4: for x being Real st x in Z holds
f . x = ((2 * a) * x) + 0 by A2;
then A5: ( sin * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * f) `| Z) . x = (2 * a) * (cos . (((2 * a) * x) + 0 )) ) ) by A3, FDIFF_4:37;
for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) )
assume A6: x in Z ; :: thesis: (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x))
then (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / (4 * a)) * (diff (sin * f),x) by A1, A5, FDIFF_1:28
.= (1 / (4 * a)) * (((sin * f) `| Z) . x) by A5, A6, FDIFF_1:def 8
.= (1 / (4 * a)) * ((2 * a) * (cos . (((2 * a) * x) + 0 ))) by A3, A4, A6, FDIFF_4:37
.= ((1 / (4 * a)) * (2 * a)) * (cos . (((2 * a) * x) + 0 ))
.= (((1 / 4) * (1 / a)) * (2 * a)) * (cos . ((2 * a) * x)) by XCMPLX_1:103
.= (((1 / 4) * 2) * ((1 / a) * a)) * (cos . ((2 * a) * x))
.= ((1 / 2) * 1) * (cos . ((2 * a) * x)) by A2, XCMPLX_1:107
.= (1 / 2) * (cos ((2 * a) * x)) by SIN_COS:def 23 ;
hence (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ; :: thesis: verum

end;

hence ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) by A1, A5, FDIFF_1:28; :: thesis: verum